Kenneth ChengMathematical modeling
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Compare the powers without decimal approximations.

Read the idea, work independently, then explain what changed.

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TOPIC 01

Exponential and logarithmic functions: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Compare the powers without decimal approximations.

(1/9)2,(1/9)3(1/9)^2,\quad(1/9)^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use exponent laws only within their real-number domain.
Hint 2
The common base is between zero and one.
Worked solution

  1. Use exponent laws only within their real-number domain.

  2. Calculate or simplify this relation.

    (1/9)2−(1/9)3=(9−1)/729>0(1/9)^2-(1/9)^3=(9-1)/729>0

  3. Increasing a positive exponent reduces a power with base between zero and one.

The squared value is larger.

Checks and common pitfalls: Increasing a positive exponent reduces a power with base between zero and one.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗